Sr Examen
Ecuación diferencial xdx+ydy+[(xdy-ydx)/(x²+y²)]=0
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
[src]
d y(x) dy*x
x + --(y(x))*y(x) - ---------- + ---------------- = 0
dx 2 2 2 2
x + y (x) dx*x + dx*y (x)
$$\frac{dy x}{dx x^{2} + dx y^{2}{\left(x \right)}} + x + y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} - \frac{y{\left(x \right)}}{x^{2} + y^{2}{\left(x \right)}} = 0$$
dy*x/(dx*x^2 + dx*y^2) + x + y*y' - y/(x^2 + y^2) = 0
Respuesta
[src]
/ / 15*dy \ / 10 6*dy \ \ / / 10 6*dy \ / 15*dy \ / 90 42*dy \ / 21*dy \ / 10 3*dy \ / 3*dy \\
| 2*|-2 - ------| 8*|-1 - --- - ------| | | 8*|-1 - --- - ------| 4*|-2 - ------| 4*|5 + --- + ------| 10*|2 + ------| 2*dy*|1 + --- + ------| 6*dy*|1 + ------||
| | 2 | | 4 2 | | | | 4 2 | | 2 | | 4 2 | | 2 | | 4 2 | | 2 ||
2 / 2 dy \ 3 / 5 4*dy \ 4 | 12 / dy \ / 10 3*dy \ \ C1 *dx/ / 3*dy \ / dy \ \ C1 C1 *dx/ 6*dy | 5 | 136 / dy \ / 10 3*dy \ \ C1 C1 *dx/ \ C1 *dx/ / 3*dy \ / dy \ / dy \ / 15*dy \ / dy \ / 30 6*dy \ \ C1 C1 *dx/ / dy \ / 10 6*dy \ / dy \ / 6*dy \ \ C1 *dx/ 90*dy \ C1 C1 *dx/ \ C1 *dx/|
x *|-1 - --- - ------| x *|1 + --- + ------| x *|--- + |-1 - ------|*|1 + --- + ------| + --------------- + 2*|1 + ------|*|-1 - ------| + --------------------- + ------| x *|-4 - --- - |-1 - ------|*|1 + --- + ------| - --------------------- - --------------- - 3*|1 + ------|*|-1 - ------| + 2*|-1 - ------|*|-2 - ------| + 2*|-1 - ------|*|-1 - --- - ------| + -------------------- + 4*|-1 - ------|*|-1 - --- - ------| + 6*|-1 - ------|*|-1 - ------| + --------------- - ------ + ----------------------- + -----------------|
| 4 2 | | 4 2 | | 4 | 2 | | 4 2 | 4 | 2 | | 2 | 4 2 | | 4 | 2 | | 4 2 | 4 4 | 2 | | 2 | | 2 | | 2 | | 2 | | 4 2 | 4 | 2 | | 4 2 | | 2 | | 2 | 4 2 2 2 |
x \ C1 C1 *dx/ \ C1 C1 *dx/ \C1 \ C1 *dx/ \ C1 C1 *dx/ C1 \ C1 *dx/ \ C1 *dx/ C1 C1 *dx/ \ C1 \ C1 *dx/ \ C1 C1 *dx/ C1 C1 \ C1 *dx/ \ C1 *dx/ \ C1 *dx/ \ C1 *dx/ \ C1 *dx/ \ C1 C1 *dx/ C1 \ C1 *dx/ \ C1 C1 *dx/ \ C1 *dx/ \ C1 *dx/ C1 C1 *dx C1 *dx C1 *dx / / 6\
y(x) = C1 + --- + ---------------------- + --------------------- + ----------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + O\x /
2 2*C1 4 3 6
C1 3*C1 24*C1 60*C1
$$y{\left(x \right)} = \frac{x^{5} \left(2 \left(-2 - \frac{15 dy}{C_{1}^{2} dx}\right) \left(-1 - \frac{dy}{C_{1}^{2} dx}\right) + 6 \left(-1 - \frac{6 dy}{C_{1}^{2} dx}\right) \left(-1 - \frac{dy}{C_{1}^{2} dx}\right) - 3 \left(-1 - \frac{dy}{C_{1}^{2} dx}\right) \left(1 + \frac{3 dy}{C_{1}^{2} dx}\right) + 2 \left(-1 - \frac{dy}{C_{1}^{2} dx}\right) \left(-1 - \frac{6 dy}{C_{1}^{2} dx} - \frac{30}{C_{1}^{4}}\right) + 4 \left(-1 - \frac{dy}{C_{1}^{2} dx}\right) \left(-1 - \frac{6 dy}{C_{1}^{2} dx} - \frac{10}{C_{1}^{4}}\right) - \left(-1 - \frac{dy}{C_{1}^{2} dx}\right) \left(1 + \frac{3 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right) - 4 + \frac{6 dy \left(1 + \frac{3 dy}{C_{1}^{2} dx}\right)}{C_{1}^{2} dx} + \frac{2 dy \left(1 + \frac{3 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right)}{C_{1}^{2} dx} - \frac{90 dy}{C_{1}^{2} dx} - \frac{4 \left(-2 - \frac{15 dy}{C_{1}^{2} dx}\right)}{C_{1}^{4}} + \frac{10 \left(2 + \frac{21 dy}{C_{1}^{2} dx}\right)}{C_{1}^{4}} - \frac{8 \left(-1 - \frac{6 dy}{C_{1}^{2} dx} - \frac{10}{C_{1}^{4}}\right)}{C_{1}^{4}} + \frac{4 \left(5 + \frac{42 dy}{C_{1}^{2} dx} + \frac{90}{C_{1}^{4}}\right)}{C_{1}^{4}} - \frac{136}{C_{1}^{4}}\right)}{60 C_{1}^{6}} + \frac{x^{3} \left(1 + \frac{4 dy}{C_{1}^{2} dx} + \frac{5}{C_{1}^{4}}\right)}{3 C_{1}^{4}} + \frac{x^{4} \left(2 \left(-1 - \frac{dy}{C_{1}^{2} dx}\right) \left(1 + \frac{3 dy}{C_{1}^{2} dx}\right) + \left(-1 - \frac{dy}{C_{1}^{2} dx}\right) \left(1 + \frac{3 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right) + \frac{6 dy}{C_{1}^{2} dx} + \frac{2 \left(-2 - \frac{15 dy}{C_{1}^{2} dx}\right)}{C_{1}^{4}} + \frac{8 \left(-1 - \frac{6 dy}{C_{1}^{2} dx} - \frac{10}{C_{1}^{4}}\right)}{C_{1}^{4}} + \frac{12}{C_{1}^{4}}\right)}{24 C_{1}^{3}} + \frac{x}{C_{1}^{2}} + \frac{x^{2} \left(-1 - \frac{dy}{C_{1}^{2} dx} - \frac{2}{C_{1}^{4}}\right)}{2 C_{1}} + C_{1} + O\left(x^{6}\right)$$
Clasificación
1st power series
lie group